The eccentricity of ellipse `4(x-2y+1)^2 +9(2x+y+2)^2=180` is

Options

A. `(sqrt(5))/(3)`
B. `(sqrt(3))/(4)`
C. `(1)/(2)`
D. `(1)/(4)`

Answer

To find the eccentricity of the given ellipse represented by the equation \( 4(x - 2y + 1)^2 + 9(2x + y + 2)^2 = 180 \), we will follow these steps:

### Step 1: Rewrite the equation in standard form
First, we need to divide the entire equation by 180 to put it in standard form.

\[ \frac{4(x - 2y + 1)^2}{180} + \frac{9(2x + y + 2)^2}{180} = 1 \]

This simplifies to:

\[ \frac{(x - 2y + 1)^2}{45} + \frac{(2x + y + 2)^2}{20} = 1 \]

### Step 2: Identify the semi-major and semi-minor axes
The standard form of an ellipse is given by:

\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]

From our equation, we can identify:

- \( a^2 = 45 \) (for the first term)
- \( b^2 = 20 \) (for the second term)

### Step 3: Calculate the eccentricity
The eccentricity \( e \) of an ellipse is given by the formula:

\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]

Substituting the values of \( a^2 \) and \( b^2 \):

\[ e = \sqrt{1 - \frac{20}{45}} = \sqrt{1 - \frac{4}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} \]

### Final Answer
The eccentricity of the ellipse is:

\[ e = \frac{\sqrt{5}}{3} \]

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